F(x)=lnx+x +bx (field x >；; 0)

F'(x)= 1/x+2x+b=(2x square+bx+1)/x.

F(x) takes the extreme value at x= 1

That is, f'( 1)=2+b+ 1=0 and b=-3.

Let f'(x)=(2x squared-3x+1)/x = 0 (x >; 0)

The solution is x1=1x2 =1/2.

So the maximum point is x= 1/2.

The minimum point is x= 1.

Second, f'(x)= 1/x+2ax+b=(2ax squared+bx+1)/x (x >; 0)

f'( 1)= 1+2a+b=0

b=- 1-2a

That is, f'(x)=[2ax squared-(2a+1) x+1]/x.

Let f'(x)=0, then (x- 1)(2ax- 1)=0.

X 1= 1, x2= 1/2a.

1) Order 1

When e> 1/2a > 1 is 2e>a>2, and f( 1) is the maximum value =a+b=- 1-a= 1, then a=-2.

When 1/2a >: =e is a> when =2e, f (1) and f (e) are the largest.

If f (1) >; F(e) then a=-2.

If f (1)

2) order1>; 2a，，a & lt 1/2

Then f( 1/2a), f(e) is the largest and f( 1) is the smallest.

f( 1/2a)& lt； F(e), a= 1/(e-2) shed

f( 1/2a)= 1